Murmurations and Sato–Tate conjectures for high rank zetas of elliptic curves

For elliptic curves over rationals, there are a well-known conjecture of Sato–Tate and a new computational guided murmuration phenomenon, for which the abelian Artin zeta functions are used. In this paper, we show that both the murmurations and the Sato–Tate conjecture stand equally well for non-abelian high rank zeta functions of the p-reductions of elliptic curves over rationals. We establish our results by carefully examining asymptotic behaviors of the p-reduction invariants $a_{E/F_p,n}(n\ge 1)$, the rank n analogous of the rank one a-invariant $a_{E/F_p}=1+p-N_{E/F_p}$ of elliptic curve $E/F_p$. Such asymptotic results are based on a ‘counting miracle’ of the so-called ${\alpha }_{E/F_q,n}$- and ${\beta }_{E/F_q,n}$-invariants of $E/F_q$ in rank n, and a remarkable recursive relation on the ${\beta }_{E/F_q,n}$-invariants, both established by Weng–Zagier in [22].